Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: Numerical analysis of a model problem.

*(English)*Zbl 0717.65086For the representation of a divergence-free vector field u defined on a bounded simply connected domain \(\Omega \subset {\mathbb{R}}^ 3\) with smooth boundary by its curl \(\omega\) and its normal component g on \(\partial \Omega\), a mixed formulation involving a vector potential \(\Psi\) is proposed. After recalling classical results on vector fields the problem is split into a homogeneous problem \((g=0)\) and a problem on the boundary (g\(\neq 0)\). Then a construction method of \(\Psi\) from the data (\(\omega\),g) is presented. The vector fields are discretized with help of finite elements. The compatibility between the two subproblems requires that the discrete field \(\Psi_ h\) has a tangential component on each point of the boundary; curved finite elements are developed that are conforming in the spaces H(div) and H(curl). They yield optimal interpolation errors. In order to guarantee uniqueness of the discrete potential \(\Psi_ h\) a special gauge condition is proposed. Optimal error estimates between the velocity u and the approximations \(u_ h\) are derived. A numerical experiment is presented (constant velocity field in a cube).

Reviewer: J.Weisel

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

76M10 | Finite element methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

35J25 | Boundary value problems for second-order elliptic equations |